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Electroweak Gauge Bosons at Future Electron-Positron Colliders

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Electroweak Gauge Bosons at Future Electron-Positron Colliders View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server LC-REV-1999-005 IKDA 99/11 June 1999 Electroweak Gauge Bosons at Future Electron-Positron Colliders Thorsten Ohl Darmstadt University of Technology Schloßgartenstr. 9 D-64289 Darmstadt, Germany <[email protected]> Abstract The interactions of electroweak gauge bosons are severely con- strained by the symmetries inferred from low energy observables. The exploration of the dynamics of electroweak symmetry breaking re- quires therefore both an experimental sensitivity and a theoretical precision that is better than the natural scale of deviations from mini- mal extrapolations of the low energy effective theory. Some techniques for obtaining improved theoretical predictions for precision tests at a + future e e− linear collider are discussed. i Preface Several excellent, up-to-date reviews of the wide range of physics that can + be probed at a future e e− Linear Collider (LC) are available [?, ?, ?]. The present review discusses some physical and technical aspects of electroweak gauge bosons, that will be important at a LC. The importance of gauge invariance even in a conservative Effective Field Theory (EFT) approach, which does not pretend to be valid to arbitrarily short distances, is em- phasized. The implementation of gauge invariance in perturbation theory is discussed. Techniques for improving the convergence and speed of high dimensional phase space integrations are described. Acknowledgments I thank Panagiotis Manakos, Hans Dahmen, and Thomas Mannel for their support over the years. Howard Georgi deserves the credit for the most inspirational year of physics so far (Howard provided an open door, physics insight, and inspiration, while DFG provided the financial support). I thank Harald Anlauf, Edward Boos, Wolfgang Kilian, and Alex Nippe for enjoyable and productive collaborations that have contributed to some of the results presented in the following pages. I thank the organizers of the Maria Laach school for the invitation to climb on the Effective Field Theory soapbox, as reproduced in part in appendix A. The students make this a fun place to lecture and the hospitality of the congregation of Maria Laach makes this school unique. I thank Deutsche Forschungsgemeinschaft (DFG) and Bundesministerium for Bildung, Wissenschaft, Forschung und Technologie (BMBF) for financial support. ii Contents 1 Introduction 1 1.1 The Art and Craft of Elementary Particle Physics . 1 1.2 Success and Failure of the Standard Model . 2 1.3TheMissionoftheLinearCollider.............. 4 1.4Roadmap............................ 7 2 The Standard Model as Effective Quantum Field Theory 9 2.1Currents............................ 10 2.2WeakInteractions....................... 13 2.3FermiTheory......................... 13 2.4 Flavor Symmetries . ...................... 15 2.5 Higher Energy Scattering . ............ 19 2.6IntermediateVectorBosons.................. 20 2.6.1 Power Counting . ............ 21 2.6.2 Hidden Symmetry . ............ 24 2.6.3 Tree Level Unitarity . ............ 26 2.7 The Minimal Standard Model . ............ 26 2.8 Precision Tests . ...................... 27 3 Matrix Elements 29 3.1 Three Roads from Actions to Matrix Elements . 29 3.1.1 ManualCalculation.................. 29 3.1.2 ComputerAidedCalculation.............. 29 3.1.3 AutomaticCalculation................. 30 3.2 Forests and Groves . ...................... 30 3.2.1 GaugeCancellations.................. 32 3.2.2 Flavor Selection Rules . ............ 34 3.2.3 Forests......................... 36 3.2.4 Flavored Forests . ............ 40 3.2.5 Groves . ...................... 40 3.2.6 Application....................... 43 iii 3.2.7 Automorphisms.................... 46 3.2.8 Advanced Combinatorics . ............ 47 3.2.9 Loops.......................... 48 4 Phase Space 49 4.1 Four Roads from Matrix Elements to Answers . 50 4.1.1 AnalyticIntegration.................. 50 4.1.2 Semianalytic Calculation . ............ 50 4.1.3 MonteCarloIntegration................ 50 4.1.4 Monte Carlo Event Generation ............ 51 4.1.5 Why We Can Not Use the Metropolis Algorithm Naively 52 4.2MonteCarlo.......................... 53 4.3 Adaptive Monte Carlo . ............ 56 4.3.1 Multi Channel . ............ 57 4.3.2 Delaunay Triangulation . ............ 59 4.3.3 VEGAS........................ 59 4.3.4 VAMP......................... 63 4.4 Parallelization . ...................... 71 4.4.1 Vegas.......................... 72 4.4.2 Formalization of Adaptive Sampling . 73 4.4.3 Multilinear Structure of the Sampling Algorithm . 75 4.4.4 Random Numbers . ............ 78 4.5DedicatedPhaseSpaceAlgorithms.............. 79 5 Linear Colliders 81 5.1TripleGaugeCouplings.................... 81 5.2 Single W ± Production..................... 85 5.3Beamstrahlung......................... 86 5.4 Quadruple Gauge Couplings and W ± Scattering....... 87 6 Conclusions 89 A The Renormalization Group 92 A.1Introduction.......................... 93 A.1.1History......................... 94 A.1.2 Renormalizability . ............ 94 A.1.3 The Renormalization Group . ............ 98 A.2 The Renormalization Group . ............ 99 A.2.1 Cracking Integrals . ............ 99 A.2.2Wilson’sInsight.................... 100 A.2.3TheSlidingCut-Off.................. 101 iv A.2.4 Renormalization Group Flow . ............ 104 A.2.5 Relevant, Marginal Or Irrelevant? . 107 A.2.6LeadingLogarithms.................. 110 A.2.7CallanSymanzikEquations.............. 110 A.2.8 Running Couplings and Anomalous Dimensions . 112 A.2.9AsymptoticFreedom.................. 114 A.2.10 Dimensional Regularization . ............ 115 A.2.11GaugeTheories.................... 116 A.3EffectiveFieldTheory..................... 118 A.3.1 Flavor Thresholds and Decoupling . 118 A.3.2 Integrating Out or Matching and Running . 120 A.3.3 Important Irrelevant Interactions . 121 A.3.4 Mass Independent Renormalization . 122 A.3.5 Naive Dimensional Analysis . ............ 123 v |1| Introduction 1.1 The Art and Craft of Elementary Particle Physics To teach how to live without certainty, and yet without being paralyzed by hesitation, is perhaps the chief thing that philosophy, in our age, can still do for those who study it. [Bertrand Russell: AHistory of Western Philosophy] In a highly idealized scenario, Theoretical Elementary Particle Physics (TEPP) proceeds by condensing the results of experimental observations up to some energy scale into a model, which is designed to remain consistent at higher energies. In a second step, predictions for experiments at these higher en- ergy scales are derived. Finally, comparisons with subsequent experimental results are used to refine the model, if necessary, and the process starts anew. Therefore, we can identify two complementary tasks: model building on one hand and calculation of observables on the other. Below, we will focus our attention in the area of model building to the systematic procedure of EFT and its rˆole in the modern interpretation of the Standard Model (SM) with simple extensions. The methods for the calculation of observables will be discussed in greater detail. Such calculations are naturally divided into two separate parts: calculation of amplitudes and differential cross sections and phase space integration. Some methods for both parts are described with a noticeable bias towards the author’s research interests. 1 Figure 1.1: Physics reaches of past, present and future colliders. Hadron colliders can not utilize the full beam energy in partonic collisions since the momentum is shared by quarks and gluons. As a rule of thumb, the lower border of the grey rectangles (√s0 = √s/10) corresponds to the physics reach of hadron colliders. (Data taken from [?, ?].) 1.2 Success and Failure of the Standard Model Tout comprendre, c’est tout mepriser. [Friedrich Nietzsche] In the decade after the commissioning of LEP1, many aspects of the Stan- dard Model [?] of elementary particle physics have been tested successfully to a precision previously unimaginable [?, ?]. Indeed it is not without irony that one of the main sources of theoretical uncertainty derives from the in- + sufficient precision of the old data on hadron production in e e−-collisions at a few GeV, despite recent theoretical progress in the interpretation of this data [?]. The important qualitative features [?] of the SM were established from low energy experiments at a time before any of the colliders shown in figure 1.1 were commissioned. The symmetry structure of the SM as a SU(2)L U(1)Y gauge theory that is spontaneously broken to the U(1) of Quantum× Elec- trodynamics (QED) has been firmly established by the stunning success of the SM at LEP1. Nevertheless, direct information regarding the underlying dynamics of the Electro Weak Symmetry Breaking (EWSB) has yet to be found. The existing data is compatible with the minimal SM with a light ele- mentary Higgs particle [?]. Global fits of the SM predictions to the LEP1 precision observables (including mtop from Tevatron and MW from LEP2) suggest a Higgs mass close to the present exclusion limits (see figure 1.2). However, the Higgs mass enters the precision observables only logarithmically and the limits are therefore weak. Despite the phenomenological hints in its favor, it must be mentioned that the minimal SM with an elementary Higgs particle is unattractive from a theoretical point of view. There is a quadratically divergent contribution 2 6 theory uncertainty ∆α(5) had = 0.02804±0.00065 0.02784±0.00026
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